Method and apparatus for concatenated channel coding with variable code rate and coding gain in a data transmission system

ABSTRACT

A novel method and apparatus for efficiently coding and decoding data in a data transmission system is described. A concatenated coding scheme is presented that is easily implemented, and that provides acceptable coding performance characteristics for use in data transmission systems. The inventive concatenated channel coding technique is well suited for small or variable size packet data transmission systems. The technique may also be adapted for use in a continuous mode data transmission system. The method and apparatus reduces the complexity, cost, size and power consumption typically associated with the prior art channel coding methods and apparatuses, while still achieving acceptable coding performance. The present invention advantageously performs concatenated channel coding without the necessity of a symbol interleaver. In addition, the present invention is simple to implement and thereby consumes much less space and power that do the prior art approaches. The present invention not only eliminates the need for a symbol interleaver between the outer and inner codes, but it also enjoys a drastically reduced implementation complexity of the inner code Viterbi decoder. The preferred embodiment of the present invention comprises an inner code having short length block codes derived from short constraint length convolutional codes utilizing trellis tailbiting and a decoder comprising four four-state Viterbi decoders having a short corresponding maximum length. The inner code preferably comprises short block codes derived from four-state (i.e., constraint length 3), nonsystematic, punctured and unpunctured convolutional code. One significant advantage of the preferred embodiment of the present concatenated coding technique is that packet data transmission systems can be designed to have variable coding gains and coding rates.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. application Ser. No.09/471,295, filed Dec. 24, 1999, entitled “Method and Apparatus forConcatenated Channel Coding in a Data Transmission System”, herebyincorporated by reference herein for its teachings on communicationsystems. This application is related to commonly-assigned U.S. Pat. No.6,016,311, issued Jan. 18, 2000 and entitled “An Adaptive Time DivisionDuplexing Method and Apparatus for Dynamic Bandwidth Allocation within aWireless Communication System”, and commonly-assigned co-pendingapplication Ser. No. 09/316,518, filed May 21, 1999 entitled “Method andApparatus for Allocating Bandwidth in a Wireless Communication System”,both references hereby incorporated by reference herein for theirteachings on communication systems.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to coding methods and apparatuses, and moreparticularly to a method and apparatus for concatenated channel codingwith variable code rate and coding gain in a data communication system.

2. Description of Related Art

As described in the commonly assigned and incorporated U.S. Pat. No.6,016,311, a wireless communication system facilitates two-waycommunication between a plurality of subscriber radio stations orsubscriber units (fixed and portable) and a fixed networkinfrastructure. Exemplary communication systems include mobile cellulartelephone systems, personal communication systems (PCS), and cordlesstelephones. The key objective of these wireless communication systems isto provide communication channels on demand between the plurality ofsubscriber units and their respective base stations in order to connecta subscriber unit user with the fixed network infrastructure (usually awire-line system). In the wireless systems having multiple accessschemes a time “frame” is used as the basic information transmissionunit. Each frame is sub-divided into a plurality of time slots. Sometime slots are used for control purposes and some for informationtransfer. Subscriber units typically communicate with a selected basestation using a “duplexing” scheme thus allowing for the exchange ofinformation in both directions of connection.

Transmissions from the base station to the subscriber unit are commonlyreferred to as “downlink” transmissions. Transmissions from thesubscriber unit to the base station are commonly referred to as “uplink”transmissions. Depending upon the design criteria of a given system, theprior art wireless communication systems have typically used either timedivision duplexing (TDD) or frequency division duplexing (FDD) methodsto facilitate the exchange of information between the base station andthe subscriber units. Both the TDD and FDD duplexing schemes are wellknown in the art.

Recently, wideband or “broadband” wireless communications networks havebeen proposed for delivery of enhanced broadband services such as voice,data and video. The broadband wireless communication system facilitatestwo-way communication between a plurality of base stations and aplurality of fixed subscriber stations or Customer Premises Equipment(CPE). One exemplary broadband wireless communication system isdescribed in the co-pending application Ser. No. 08/974,376 and is shownin the block diagram of FIG. 1. As shown in FIG. 1, an exemplarybroadband wireless communication system 100 includes a plurality ofcells 102. Each cell 102 contains an associated cell site 104 thatprimarily includes a base station 106 and an active antenna array 108.Each cell 102 provides wireless connectivity between the cell's basestation 106 and a plurality of customer premises equipment (CPE) 110positioned at fixed customer sites 112 throughout the coverage area ofthe cell 102. The users of the system 100 may include both residentialand business customers. Consequently, the users of the system havedifferent and varying usage and bandwidth requirement needs. Each cellmay service several hundred or more residential and business CPEs.

The broadband wireless communication system 100 of FIG. 1 provides true“bandwidth-on-demand” to the plurality of CPEs 110. CPEs 110 requestbandwidth allocations from their respective base stations 106 based uponthe type and quality of services requested by the customers served bythe CPEs. Different broadband services have different bandwidth andlatency requirements. The type and quality of services available to thecustomers are variable and selectable. The base station media accesscontrol (“MAC”) allocates available bandwidth on a physical channel onthe uplink and the downlink. Within the uplink and downlink sub-frames,the base station MAC allocates the available bandwidth between thevarious services depending upon the priorities and rules imposed bytheir quality of service (“QoS”). The MAC transports data between a MAC“layer” (information higher layers such as TCP/IP) and a “physicallayer” (information on the physical channel).

Due to several well known communication phenomenon occurring in thetransmission link between the base stations 106 and the CPEs 112, it iswell known that the transmission links or channels may be noisy andthereby produce errors during transmission. These errors are typicallymeasured as Bit Error Rates (BERs) that are produced during datatransmission. Depending upon the severity of these errors, communicationbetween the base stations 106 and the CPEs 112 can be detrimentallyaffected. As is well known, by properly encoding data, errors introducedby noisy channels can be reduced to any desired level withoutsacrificing the rate of information transmission or storage. SinceShannon first demonstrated this concept in his landmark 1948 paperentitled “A Mathematical Theory of Communication”, by C. E. Shannon,published in the Bell System Technical Journal, pps. 379-423 (Part I),623-656(Part II), in July 1948, a great deal of effort has been putforth on devising efficient coding and encoding methods for errorcontrol in a noisy communication environment. Consequently, use of errorcorrecting coding schemes has become an integral part in the design ofmodern communication systems.

For example, in order to compensate for the detrimental effects producedby noisy communication channels (or for noise that may be generated atboth the sources and destinations), the data exchanged between the basestations 106 and the CPEs 112 of the system 100 of FIG. 1 may be codedusing conventional combined coding and modulation designs. For example,convolutional or trellis-coded modulation (TCM)-Reed-Solomon (RS) typecoders are well known in the art and can be used to code the data as itis exchanged in the system 100 of FIG. 6. Convolutional or TCM-RSconcatenation coding schemes are well known in the communication art asexemplified by their description in the text entitled “ConvolutionalCoding, Fundamentals and Applications”, by L. H. Charles Lee, publishedby Artech House, Inc. in 1997, the entire text of which is hereby fullyincorporated by reference for its teachings on convolutional/TCM-RScoding schemes and techniques. As is well known, in the past, channelcoding designs and modulation designs were treated as separate entities.Hamming distance was considered an appropriate measure for systemdesign. TCM design offers the optimum matching between the channelencoder output code vector and the modulator using a special signalmapping technique.

As is well known, the coding gains produced by coding schemes employingconvolutional or TCM coding schemes for the inner codes and RS for theouter codes (i.e., concatenating the convolutional/TCM inner codes withthe RS outer codes) is relatively high in terms of the minimum Hammingdistance and coding rates achieved. Disadvantageously, the high codinggains achieved by these conventional schemes come at a price in terms ofcomplexity, cost, size, speed, data transmission delays and power.

As is well known to those of skill in the art, one of the maindisadvantages associated with the prior art concatenated coding schemesis that these techniques require the use of symbol “interleavers”. TheConvolutional/TCM-RS concatenation techniques must employ a symbolinterleaver between the outer and inner codes because when the innercode decoder makes a decoding error, it usually produces a long burst oferrors that affect multiple consecutive symbols of the outer decoder.Thus without a deinterleaver, the performance of the outer decoderseverely degrades and the effective coding gains produced by theconcatenation is lost. Furthermore, the presence ofinterleaver/deinterleaver distributes the error bursts over multipleouter code words thereby effectively utilizing the power of the outercodes. In communication systems that transmit only short or variablelength packets, a symbol interleaver cannot be utilized because it isimpractical. In addition, the symbol interleaver required by the priorart concatenated channel coding schemes increase delays in datatransmission.

These increased transmission delays may be unacceptable in someapplications. For example, when the system 100 of FIG. 1 is used tocommunicate T1-type continuous data services between the base stationsand the CPEs. These type of data services often have well-controlleddelivery latency requirements that may not tolerate the transmissiondelays introduced by the symbol interleavers utilized by theconcatenated channel coding schemes of the prior art.

A second disadvantage associated with the prior art concatenated codingschemes is that these techniques require additional overhead for trellistermination at the end of packets. This additional overhead lessens theeffective coding gains obtained via concatenation. A third disadvantageassociated with the prior art concatenated coding schemes is the use ofViterbi decoders for constraint length 7 codes. As is well known tothose of skill in the art, constraint length 7 Viterbi decoders causesignificant decoding delays that are undesirable for packet datatransmissions. Furthermore, these decoders are relatively complex andtherefore must be implemented using devices having increased cost andpower characteristics.

The prior art concatenated channel coding schemes are relatively complexto implement and therefore suffer the power, size, and reliabilitydisadvantages as compared with less complex implementations. As aresult, prior art channel coding implementations for packet datatransmission systems have typically used “single level” codingtechniques such as convolutional, TCM or block coding techniques.Examples of block codes are Bose-Chaudhuri-Hocquenghem (BCH) codes,Reed-Muller (RM) codes, cyclic codes, array codes,single-error-correcting (SEC) Hamming codes, and Reed-Solomon (RS)codes. Therefore, disadvantageously, packet transmission systems, havenot heretofore been able to take advantage of the benefits offered byconventional concatenation coding techniques that provide the advantageof soft-decision decoding of the inner code resulting in larger codinggain and better coding efficiencies.

Therefore, a need exists for a concatenated channel coding method andapparatus that can be easily implemented, provides acceptable codingperformance, is well suited for use in small or variable size packetdata transmission systems, and overcomes the disadvantages of the priorart concatenated channel coding methods and apparatuses. Specifically, aneed exists for a concatenated channel coding method and apparatus thatrequires no additional overhead caused by trellis termination andreduces implementation complexity and decoding delays. Furthermore, aneed exists for a concatenated channel coding method and apparatus forpacket data transmission that allows for varying coding gain. Thepresent invention provides such a concatenated coding method andapparatus.

SUMMARY OF THE INVENTION

The present invention is a novel method and apparatus for efficientlycoding data in a data transmission system. The inventive concatenatedchannel coding technique is well suited for small or variable sizepacket data transmission systems. The technique may also be adapted foruse in a continuous mode data transmission system. The method andapparatus reduces the complexity, cost, size and power consumptiontypically associated with the prior art channel coding methods andapparatuses, while still achieving acceptable coding performance. Thepresent invention advantageously performs concatenated channel codingwithout the necessity of a symbol interleaver. In addition, the presentinvention is simple to implement and thereby consumes much less spaceand power that do the prior art approaches. The present invention notonly eliminates the need for a symbol interleaver between the outer andinner codes, but it also enjoys a drastically reduced implementationcomplexity of the inner code Viterbi decoder.

The inventive concatenation technique does not require a symbolinterleaver (or deinterleaver on the decoder end) because when the innercode makes a decoding error, it produces only single or a limited numberof outer code symbol errors. The present method and apparatus eithercorrects for the noisy received symbol using a soft decision decodingtechnique or it produces the erroneous symbol on the output.Consequently, the inner code can be considered as being completelymatched or in other words completely dedicated to the task of assistingthe outer code in working best.

The asymptotic coding gain of a code decoded with optimum decoding isgiven as 10log₁₀(r d_(min)), where r is the code rate and d_(min) is theminimum Hamming distance of the code. The convolutional/TCM codeemployed in the conventional concatenated coding usually use an innercode having larger d_(min) but the code rate is usually low. The higherthe d_(min), the more complex the code usually is. In accordance withthe present inventive coding technique, an inner code is selected tohave a relatively modest d_(min) value. However, the coding rate isimproved and better than the code used by the conventional prior artconcatenated coding schemes. Another important parameter which hasaffect on the performance is N_(dmin). This is the number of paths atdistance d_(min) from the correct path. Low values of N_(dmin) aredesirable for better performance. But usually, the higher the d_(min),the more complex the code is to implement and it also has lower rate andhigher N_(dmin).

The inner codes used by the present inventive coding technique have thefollowing three advantages as compared to the prior art approaches: (1)the inner code is matched to the requirements and characteristics of theouter code (this assists the outer decoder in decoding the code in anoptimum manner; (2) the inner code yields a coding technique havingmoderately high coding rates thereby providing good coding gains withmodest d_(min) values; and (3) the inner code yields low values ofN_(dmin).

One preferred embodiment of the present invention comprises an innercode having short length block codes derived from short constraintlength convolutional codes. The present invention utilizes trellistailbiting and a decoder comprising four four-state Viterbi decodershaving a short corresponding maximum length. The inner code preferablycomprises short block codes derived from a four-state (i.e., aconstraint length of 3), nonsystematic, punctured and unpuncturedconvolutional code. The inner code also preferably utilizes trellistailbiting techniques. One significant advantage of the preferredembodiment of the present concatenated coding technique is that packetdata transmission systems can be designed to have variable coding gainsand coding rates.

In the preferred embodiment of the present invention, the outer code isa (N,K) Reed-Solomon code over GF (2^(m)). The inner code preferably isa (m+1, m) parity-check code. The minimum Hamming distance d_(min) ofthe inner code is 2. The overall code rate is given by the followingequation (Equation 1): $\begin{matrix}{{r = {\frac{Km}{N\quad\left( {m + 1} \right)} = \frac{Km}{\left( {K + R} \right)\quad\left( {m + 1} \right)}}};} & {{Equation}\quad 1}\end{matrix}$where R is the redundancy of the RS code, N is the length (measured insymbols) of the RS code, K is the message length (in symbols), and m isthe length of the symbol in bits.

The single parity can be computed in parallel by an exclusive-OR ofm-input bit. Alternatively, the single parity bit can be computed in asequential manner with a single shift register and a single exclusive-ORgate.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a simplified block diagram of a broadband wirelesscommunication system adapted for use with the present invention.

FIG. 2 is a simplified block diagram of an encoder made in accordancewith the present invention.

FIG. 3 is a simplified a block diagram of a decoder made in accordancewith the present inventive concatenation channel coding method andapparatus.

FIG. 4 shows a Trellis diagram of the parity check code in accordancewith the present concatenated channel coding technique.

FIG. 5 depicts a graph showing the performance of a data transmissionsystem using QPSK modulation and the concatenated channel codingtechnique of the present invention.

FIG. 6 depicts a graph showing the BER performance characteristics of afirst exemplary embodiment of the present invention.

FIG. 7 depicts a graph showing the BER performance characteristics of asecond exemplary embodiment of the present invention.

FIG. 8 depicts a graph showing the BER performance characteristics of athird exemplary embodiment of the present invention.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION OF THE INVENTION

Throughout this description, the preferred embodiment and examples shownshould be considered as exemplars, rather than as limitations on thepresent invention.

One significant advantage of the present concatenated coding techniqueis that it not only eliminates the need for a symbol interleaver betweenthe outer and inner codes, but it also enjoys drastically reducedimplementation complexity of the inner code Viterbi decoder. That is,the Viterbi decoder used to implement the inner code is much lesscomplex than those required by the prior art approaches. Viterbicoders/decoders are well known in the art and are described in much moredetail in a text by Shu Lin and Daniel Costello, Jr., entitled “ErrorControl Coding, Fundamentals and Applications”, published by PrenticeHall in 1983, the entire text of which is hereby incorporated byreference herein for its teachings on error control coding.

The present inventive concatenation technique does not require use of aninterleaver/deinterleaver because when the inner code makes a decodingerror it produces either only a single or a few outer code symbolerrors. In accordance with the present invention, the decoder either“cleans” a noisy received symbol with the aid of soft decision decodingmethod, or it generates the erroneous symbol. Thus, the inner code inthe new system can be considered completely matched, or, in other words,completely dedicated to the task of aiding the performance of the outercode.

As is well known, the asymptotic coding gain of a code decoded withoptimum decoding is given as 10log₁₀(r d_(min)), where r is the coderate and d_(min) is the minimum Hamming distance of the code. The priorart convolutional/TCM code employed in the prior art concatenated codersusually use an inner code having larger d_(min) values. However, thecode rate associated with these coders is usually low. The higher thed_(min) value, the more complex is the code. In the concatenated codingscheme presented herein, an inner code is selected having a modestd_(min) value. However, the coding rate is superior to the code used inconventional concatenated code schemes. Also, as is well known, anotherimportant parameter which has affect on the performance is N_(dmin).This is the number of paths at distance d_(min) from the correct path.Low values of N_(dmin) are desirable for better performance. Usually,however, the higher the d_(min) value is, the more complex is the codeto implement. In addition, it also has a lower rate and a higherN_(dmin) value.

The strength of the inner code used in the inventive concatenated codingtechnique can be summarized as follows: (1) the inner code is matched tothe needs and characteristics of the outer code, thus helping the outerdecoder in an optimum manner; (2) the inner codes have medium to highcoding rates thus providing acceptable coding gain even with modestd_(min); and (3) the inner codes yield relatively low values ofN_(dmin). Note that for m=8, the asymptotic coding gain of the parityinner code is 10log10(2*8/9)=2.49 dB.

FIG. 2 shows a simplified block diagram of an encoder made in accordancewith the present invention. As shown in FIG. 2, the inventive encoderpreferably comprises a concatenated channel encoder 200 having an outercode encoder 202 operatively coupled to an inner code encoder 204. Theouter code encoder preferably uses an (N,K) Reed-Solomon code over GF(2^(m)). These types of RS codes are well known in the art and can beimplemented using the teachings provided in the text by Lin andCostello, fully incorporated herein as stated above. In accordance withthe present invention, the inner encoder 204 preferably uses an innercode comprising an (m+1,m) parity-check code. Preferably, the minimumHamming distance d_(min), of the inner code is 2. The overall code rater is given by the Equation 1: $\begin{matrix}{r = {\frac{Km}{N\quad\left( {m + 1} \right)} = \frac{Km}{\left( {K + R} \right)\quad\left( {m + 1} \right)}}} & {{Equation}\quad 1}\end{matrix}$where, as stated above, R is the redundancy of the RS code; N is thelength (measured in symbols) of the RS code, K is the message length (insymbols), and m is the length of the symbol in bits. The single paritybit can be computed in parallel using an exclusive-OR of m-input bitcircuit. Alternatively, the single parity bit can be computed in asequential manner with a single shift register and a single exclusive-ORgate in a well-known manner.

FIG. 3 shows a block diagram of a decoder made in accordance with thepresent inventive concatenation channel coding method and apparatus. Asshown in FIG. 3, the inventive decoder 300 preferably comprises aMaximum likelihood “soft decision” parity check code decoder 302operatively coupled to an error-only or error and erasure RS codedecoder 304. The parity check code decoder 302 accepts “soft channelbits” in a well known fashion from the communication channel and thedemodulator. In the embodiment shown, the soft channel bits comprise“m+1” bits, while the input to the RS decoder 304 comprises “m” bits.The decoder is preferably implemented using a relatively non-complexsoft-decision Viterbi decoder which is well known in the art. Suchdecoders are described in detail at pages 315-384 of the Lin andCostello reference which is incorporated by reference hereinabove.

FIG. 4 shows a Trellis diagram 400 of the (m+1,m) parity check code inaccordance with the present concatenated channel coding technique. Asshown in FIG. 4, the inner code has 2-states trellis diagram with (m+1)stages in the trellis. Hence it can be decoded using a very simple andstraightforward soft-decision Viterbi decoder. The trellis terminates tozero state in (m+1) stages. Note that since there are only two states inthe trellis and length of the trellis is a small number m (typical valuefor m is “8”), a register exchange method can be utilized for storingthe decoded path through the trellis. Thus, advantageously, there is nodecoding delay through this simple 2-states Viterbi decoder. Incomparison, disadvantageously, the Viterbi decoder used by the prior artconcatenated coding schemes employing convolutional/TCM codes is muchmore complex and has significant decoding delays.

In an alternative embodiment, the inner code can be decoded by (1)performing a correlation of the received vector of length “m” with 2^(m)possible code words, and (2) selecting as the decoded output the codeword that has a maximum correlation metric. This is a relatively“brute-force” approach for performing optimum decoding. The trellis is amuch more efficient way to perform the same task. Alternatively, if theinput bit rate is relatively high, it may be desirable to processmultiple input samples in parallel. The trellis diagram of FIG. 4 caneasily be modified to implement this modified approach. This results ina trellis having an increased number of branches but of shorter length.

For example, if the inner code is (9,8) code, then it is possible tocombine the three stages in the trellis into a single stage and processthree input bits at a time. These are only a few examples of possiblealternative implementations possible for the inner code decoder.Essentially these are all equivalent implementations. As one of ordinaryskill in the art shall recognize, many alternative implementationapproaches may be used without departing from the scope of the presentinvention.

The inner code works with all types of signal constellations. For higherlevel constellations such as 16 QAM, multiple branch metrics arecomputed for a single received I,Q pair as follows.

For example, let (b3, b2, b1, b0) represent the 4-bit binary label of a16 QAM signal constellation. Let y denote the received signal point andS denote the set of 16 QAM signal points for which b3=0. Then the branchmetric for bit b3 for hypothesis 0 is given by:$\min\limits_{x \Subset S}\quad{{y - x}}^{2}$

Similarly the metrics for other hypothesis and other bits are computed.It is possible to use the distance rather than the squared distance forthe branch metrics. A bit permutator between the inner encoder andmodulation symbol mapper may be employed to make the branch metrics inthe consecutive stages of the trellis uncorrelated for higher levelmodulations. This approach may slightly improve the concatenated codeperformance.

Performance Characteristics using the Present Inventive Channel CodingTechniques

Performance characteristics of the inventive coding method and apparatusare now provided. The performance characteristics of an error-only RSdecoding technique are described.

The error event probability of the inner code with BPSK or QPSKmodulation and decoded with maximum-likelihood soft decision Viterbidecoding is given by the following upper bound (Equation 2):$\begin{matrix}{{P_{e} < {\sum\limits_{d = d_{m\quad i\quad n}}^{\infty}\quad{a_{d}\quad Q\quad\left( \sqrt{2{{rdE}_{b}/N_{0}}} \right)}}};} & {{Equation}\quad 2}\end{matrix}$where, a_(d) is the number of incorrect paths at Hamming distance d fromthe correct path that diverge from the correct path and remerge to it atsome later stage. One of ordinary skill in the coding/decoding art shallrecognize that from the trellis diagram of FIG. 4, for${\left( {{m + 1},m} \right)\quad{parity}\quad{check}\quad{code}\quad a_{dmin}} = {{\left( \underset{1}{m} \right) + {\left( \underset{2}{m} \right)\quad{where}\quad\left( \underset{k}{m} \right)}} = {\frac{m!}{{k!}\quad{\left( {m - k} \right)!}}.}}$

At higher SNR, only the first term is significant. Ignoring the higherorder terms convert the above bound (of Equation 2) into an approximateexpression. Hence the probability of symbol error at the input of the RSdecoder can be given by the following expression (Equation 3):$\begin{matrix}{P_{s} \approx {\left\lbrack {\left( \underset{1}{m} \right) + \left( \underset{2}{m} \right)} \right\rbrack\quad Q\quad\left( \sqrt{4{{rE}_{b}/N_{0}}} \right)}} & {{Equation}\quad 3}\end{matrix}$

The block error probability of the RS code with redundancy R is given bythe following equation (Equation 4): $\begin{matrix}{P_{block} = {\sum\limits_{i = {{R/2} + 1}}^{N}\quad{\left( \underset{i}{N} \right)\quad\left( P_{s} \right)^{i}\quad\left( {1 - P_{s}} \right)^{N - i}}}} & {{Equation}\quad 4}\end{matrix}$

Once again, at higher SNR, only the first term in the summation givenabove is significant. The bit error probability at the output of RSdecoder is approximately given by the following expression (Equation 5):$\begin{matrix}{P_{b} \approx {2\quad\frac{\frac{R}{2} + 1}{Nm}\quad\left( \underset{\frac{R}{2} + 1}{N} \right)\quad\left( P_{s} \right)^{\frac{R}{2} + 1}\quad\left( {1 - P_{s}} \right)^{N - {({\frac{R}{2} + 1})}}}} & {{Equation}\quad 5}\end{matrix}$

An example is now described. Let K=54 bytes, thus m=8. FIG. 5 depicts agraph showing the performance of a data transmission system using QPSKmodulation and the concatenated channel coding technique of the presentinvention. The bit error rate of this concatenated system is shown inFIG. 5 for R=6, 8, 10 and 12. At 10⁻⁹ output BER, the coding gain andoverall code rate for these 4 codes are presented in Table 1 below.

In one alternative embodiment, the Viterbi decoder of FIG. 3 can bemodified such that it outputs reliability information for each symbolthat is generated. An error and erasure correcting RS decoder can thenbe used to further improve the performance of the system using thepresent invention.

TABLE 1 The Code Rate and Gain for the Exemplary system using InventiveConcatenated Coding Scheme Code Coding Gain R Rate (dB)  6 0.80 5.30  80.774 5.65 10 0.75 5.90 12 0.7272 6.10Concatenated Coding Techniques having Variable Coding Gains and CodingRates

One significant advantage of the preferred embodiment of the presentconcatenated coding technique is that packet data transmission systemscan be designed to have variable coding gains and rates. Thus, adesigner of such a system can choose to design the system to have ahigher coding gain at a cost of a lower coding efficiency, and viceversa. Another advantage of the present invention is the elimination ofthe need for a symbol interleaver between the outer and inner codes.

The preferred embodiment of the present invention offers the same codingrate flexibility as a standard RS/convolutional concatenated code whileproducing similar or better coding gain. Several advantages of utilizingthe preferred embodiment of the present invention instead of thestandard RS/convolutional concatenated code are now discussed. First,the present invention utilizes trellis tailbiting, which requires noadditional overhead for trellis termination. Disadvantageously, thestandard code having a constraint length 7 code requires an additionaloverhead of 6 bits at the end of each packet for trellis termination.Second, the inner code of the present invention enjoys drasticallyreduced implementation complexity than a constraint length 7 standardcode. The present inner code decoder is only slightly more complicatedthan a parity check decoder. Third, the inner code of the presentinvention enjoys a much smaller decoding delay than the delays observedusing a constraint length 7 standard code, and thus, provides awell-suited mechanism for packet data transmission. Fourth, as describedabove, the present invention does not require use of an interleaver;however, the present invention can utilize a relatively small sizeinterleaver and provide additional coding gain. For example, a depth 4interleaver provides an additional 0.5 to 1.0 dB of coding gain. Fifth,the present invention can be used for both uplink and downlinkcommunication channels without interleaving, and thus provides equalcoding gains for both uplink and downlink. One preferred embodiment ofthe present invention is now described.

The preferred embodiment of the present invention comprises an innercode having short length block codes derived from short constraintlength convolutional codes utilizing trellis tailbiting and a decodercomprising 2^(k−1) 2^(k−1) states Viterbi decoders having a shortcorresponding maximum length, where “k” is the constraint length of thecode. The inner code preferably comprises short block codes derived fromfour-state (i.e., constraint length 3), nonsystematic, punctured andunpunctured convolutional codes. Punctured convolutional codes are wellknown in the art and one such technique of providing puncturedconvolutional codes is described in detail in U.S. Pat. No. 5,511,082issued to How et al., on Apr. 23, 1996, entitled “PuncturedConvolutional Encoder”, the entire reference hereby incorporated byreference herein for its teachings on communication encoding. Thegenerator polynomials of the inner code are preferably 7 and 5. Theinner code preferably utilizes trellis tailbiting techniques (e.g.,utilizing the last K-1 bits of a block to initialize encoder memory,where K is the constraint length). Trellis tailbiting techniques arewell known in the art. One such technique is described in detail in anarticle by Jack Keil Wolf and Andrew J. Viterbi, entitled “On the WeightDistribution of Linear Block Codes Formed From Convolutional Codes”,published by IEEE in the IEEE Transactions on Communications Vol. 44,No. 9, September 1996, the entire text of which is hereby incorporatedby reference herein for its teachings on coding techniques.

The present invention preferably utilizes trellis tailbiting (instead ofutilizing trellis termination) because the use of trellis terminationfor short length block codes results in excessive rate loss. The presentinvention preferably utilizes a relatively short constraint length(e.g., 3 or 4) for the following reasons related to trellis tailbiting.As is well known to those skilled in the coding art, trellis tailbitingtechniques generate block code lengths of a particular minimum size toobtain a weight distribution that is similar to the weight distributionobtained using trellis termination. The larger the constraint length is,the longer the block code length needs to be. This causes degradedperformance of the outer codes because the inner codes generatecorrelated multiple symbol errors. The inner codes generate correlatedsymbol errors because the inner code message lengths usually do notmatch the outer code symbol sizes. This mismatch effect is morepronounced for higher constraint length codes. Trellis tailbitingtechniques implemented for maximum likelihood decoding require 2^(k−1)trellises. This implies that trellis tailbiting is impractical for longconstraint lengths. Thus, short constraint lengths of 3 or 4 arepreferably used in the present invention. Three exemplary embodiments ofthe present invention are now described in detail.

A First Exemplary Embodiment—Inner Code K=3, Rate ½ Convolutional Code

A first exemplary embodiment comprises an inner code having a shortlength block code derived from an unpunctured rate ½ constraint length 3convolutional code. For each inner code message block, trellistailbiting requires that the encoder memory be initialized with the lasttwo bits of the block. In the first exemplary embodiment the messagelength for the inner code is preferably equal to or greater than 8 bits.For outer RS codes based on GF(2^(n)), the inner code message length inbits is preferably a multiple of n. Thus, for an outer RS code based onGF(2⁸), the inner code message length (in bits) is preferably a multipleof 8. Similarly, for an outer RS code based on GF(2⁹) or GF(2¹⁰) theinner code message length (in bits) is preferably a multiple of 9 or 10,respectively. These configurations are exemplary only as those skilledin the coding art shall recognize that other configurations can be usedwith the present invention without departing from the scope and spiritof the present invention. For example, inner codes can be derived fromhigher constraint lengths, however, decoding complexity increasesgreatly and the minimum message length for the inner code alsoincreases.

A Second Exemplary Embodiment—Inner Code K=3, Rate ⅔ Convolutional Code

A second exemplary embodiment comprises an inner code having a shortlength block code derived from a punctured rate ⅔ constraint length 3convolutional code. For each inner code message block, trellistailbiting requires that the encoder memory be initialized with the lasttwo bits of the block. The punctured pattern is preferably 11 and 10corresponding to generator polynomials 7 and 5, respectively. Thispunctured pattern is exemplary only as one skilled in the art shallrecognize that other punctured patterns can be used with the presentinvention without departing from the spirit of the invention. The innercode length of the second exemplary embodiment is preferably equal to orgreater than 8 bits and a multiple of the puncture period (i.e., 2). Asdescribed above with reference to the first exemplary embodiment, forouter RS codes based on GF(2⁸), the inner code message length (in bits)is preferably a multiple of 8. Thus, advantageously, the same inner codedecoder can be used for both the first and the second exemplaryembodiments.

A Third Exemplary Embodiment—Inner Code K=3. Rate ⅘ Convolutional Code

A third exemplary embodiment of the present invention comprises an innercode having a short length block code derived from a punctured rate ⅘constraint length 3 convolutional code. For each inner code messageblock, trellis tailbiting requires that the encoder memory beinitialized with the last two bits of the block. The punctured patternis preferably 1001 and 1110, corresponding to generator polynomials 7and 5, respectively. This punctured pattern is exemplary only as oneskilled in the art shall recognize that other punctured patterns can beused with the present invention without departing from the spirit of theinvention. The inner code length of the third exemplary embodiment ispreferably equal to or greater than 16 bits and a multiple of thepuncture period (i.e., 4). As described above with reference to thefirst and second exemplary embodiments, for outer RS codes based onGF(2⁸) the inner code message length in bits is preferably a multiple of8. Thus, advantageously, the same inner code decoder can be used for allthree exemplary embodiments of the present invention.

Performance Characteristics of the Exemplary Embodiments of the PresentInventive Channel Coding Technique

Performance characteristics of the exemplary embodiments of theinventive coding method and apparatus are now provided. BER performancecharacteristics of the above-described exemplary embodiments of thepresent inventive concatenated coding method and apparatus arecalculated using a semi-analytical method. The semi-analytical method isdescribed hereinbelow for an inner code message length of 32 bits and aGF(2⁸)-based RS outer code. One skilled in the art shall recognize thatdifferent inner code message lengths and RS symbol sizes can be utilizedwith the present semi-analytical method to provide performancecharacteristics of the present invention.

The bit error probability (P_(b)) is the probability of a bit erroroccurring at the output of the inner code decoder. Similarly, the byteerror probability (P_(byte)) is the probability of a byte erroroccurring at the output of the inner code decoder. P_(w), P_(x), P_(y),and P_(z), represent the probabilities of a 32-bit decoded word having1, 2, 3 and 4 byte errors, respectively, occurring at the output of theinner code decoder. The BERs at the output of a t-byte correcting RSdecoder ((P_(e))_(RS)) can be given by the following expression(Equation 6), where N=(M+2t)/4 and M is the message size in bytes:$\begin{matrix}{{\left( P_{e} \right)_{RS} = {\frac{P_{b}}{P_{byte}}\quad{\sum\limits_{\substack{{w + {2x} + {3y} + {4z}} > t \\ {w + x + y + z} \leq N \\ {0 \leq w},x,y,{z \leq N}}}^{\quad}\quad{\frac{w + {2x} + {3y} + {4z}}{4N}\quad\left( \underset{w}{N} \right)\quad\left( \underset{x}{N - w} \right)\quad\left( \underset{y}{N - w - x} \right)\quad\left( \underset{z}{N - w - x - y} \right)}}}}\quad{P_{w}^{w}\quad P_{x}^{x}\quad P_{y}^{y}\quad P_{z}^{z}\quad\left( {1 - P_{w} - P_{x} - P_{y} - P_{z}} \right)^{N - w - x - y - z}}} & {{Equation}\quad 6}\end{matrix}$

Those skilled in the coding arts shall recognize that the probabilitiesP_(b), P_(byte), P_(w) P_(x), P_(y) and P_(z) can be obtained fromsimulations for different Eb/No. Preferably at least 500 error events ofeach type are simulated to ensure the accuracy of the probabilitiesobtained via simulation. Equation 6 is preferably evaluated using acomputer or similar device because of the large number of terms withinthe equation. The BER performance characteristics of the above-describedexemplary embodiments are now described with reference to FIGS. 6-8.

FIG. 6 depicts a graph showing the BER performance characteristics ofthe first exemplary embodiment of the present invention, a concatenatedcode having an inner code based on (8,4) extended Hamming code and aconcatenated code having an inner code based on (24, 12) extended Golaycode. The BER performance of the first exemplary embodiment of thepresent invention shown in FIG. 6 is for a message block length of 56bytes, RS code redundancy of 16 bytes and inner code message length of 8bits. The overall code rate of this exemplary embodiment is 0.38.

For comparison purposes, FIG. 6 depicts two concatenated codes havinginner codes based on an (8,4) extended Hamming code and a (24, 12)extended Golay code. If these codes are decoded with trellis-basedmaximum likelihood decoding algorithms, they will generally have highercomplexity trellises (either more states and/or time varying irregulartrellises) than the present invention. Thus, utilization of the Hammingcode and Golay code result in a more complex inner code decoder. TheHamming code utilizes a GF(2⁸) based RS code and the Golay code utilizesa GF(2¹²) based RS code.

As shown in FIG. 6, the new concatenated code outperforms theHamming-based concatenated code by 0.75 dB. The powerful Golay-basedconcatenated code only approaches the performance of the first exemplaryembodiment at an output BER of 10⁻⁹. As is well known, theimplementation complexity of an (8,4) Hamming code is comparable orslightly higher than the inner code of the first exemplary embodiment,whereas disadvantageously, the implementation complexity of the maximumlikelihood Golay code decoder is an order of magnitude higher. Oneskilled in the art shall recognize that an RS/constraint length 7 rate ½concatenated code without interleaving has the same performance as thefirst exemplary embodiment, but at the cost of much higher decodingcomplexity and decoding delay.

FIG. 7 depicts a graph showing the BER performance characteristics ofthe second exemplary embodiment of the present invention having an innercode message length of 16 bits, an inner code message length of 32 bitsand a concatenated code having an inner code based on (12,8) shortenedHamming code. The BER performance characteristics of the secondexemplary embodiment of the present invention shown in FIG. 7 are for amessage block length of 56 bytes, RS code redundancy of 16 bytes andinner code message length of 16 and 32 bits. The overall code rate ofthis exemplary embodiment is 0.51.

As shown in FIG. 7, the second exemplary embodiment outperforms the morecomplex Hamming-based concatenated code system for BER outputs up to10⁻⁹, which, as is well known, is a typical operating region for highdata-rate/high-performance systems. One skilled in the art shallrecognize that, disadvantageously, a (12,8) Hamming code decoder isnecessarily different from a (8,4) Hamming code decoder. Thus,implementing both Hamming code rates requires implementation of aseparate decoder for each rate. Advantageously, the present inventionutilizes one decoder for all rates.

FIG. 8 depicts a graph showing the BER performance characteristics ofthe third exemplary embodiment of the present invention having an innercode message length of 32 bits. The BER performance characteristics ofthe third exemplary embodiment of the present invention shown in FIG. 8are for a message block length of 212 bytes, RS code redundancy of 32bytes and inner code message length of 32 bits. The overall code rate ofthis exemplary embodiment is 0.71.

Summary:

A novel inventive concatenated coding scheme has been described, whereinthe outer code comprises an RS code over GF(2^(m)) and the inner codecomprises a (m+1,m) single parity check code. The inner code ispreferably decoded using maximum likelihood soft-decision decoding suchas is performed using a Viterbi decoding method. In one embodiment,information is provided to the outer decoder regarding the reliabilityof the symbol that is decoded. The outer decoder preferably compriseseither an error-only or error and erasure correcting RS decoder. Thepreferred embodiment of the present invention comprises an inner codehaving short length block codes derived from short constraint lengthconvolutional codes utilizing trellis tailbiting and a decodercomprising four four-state Viterbi decoders having a short correspondingmaximum length. The inner code preferably comprises short block codesderived from four-state (i.e., constraint length 3), nonsystematic,punctured and unpunctured convolutional code. The inner code alsopreferably utilizes trellis tailbiting techniques.

One significant advantage of the preferred embodiment of the presentconcatenated coding technique is that packet data transmission systemscan be designed to have variable coding gains and coding rates. Anotheradvantage of the present invention is that the need for a symbolinterleaves between the outer and inner codes is eliminated. Thepreferred embodiment of the present invention offers the same codingrate flexibility as a standard RS/convolutional concatenated code whileproducing similar or better coding gains.

Three exemplary concatenated coding schemes have been described. Theseschemes are well suited for packet data transmission and also have muchsimpler implementation complexity than the prior art schemes. The codingschemes of the present invention are remarkably simple to implement(requires only a small number of states Viterbi decoder of short length)and provide variable coding gains with variable code rates. Theexemplary decoder consists of four four-state Viterbi decoders of amaximum length of 32. Implementation of the exemplary decoder is verysimple because it does not require traceback memory, tracebackmechanisms, path metrics normalization, etc. that are required in theimplementation of a typical Viterbi decoder that has a larger constraintlength and a larger block length. All three exemplary schemes share thesame decoder for decoding the inner codes. The three exemplary schemesprovide the same or better performance than provided by prior arttechniques yet at a drastically reduced implementation complexity.

A further performance improvement is realized when interleaving of shortdepth (e.g. 2 or 4) is used for continuous data transmission systems.Thus, the present invention can be effectively utilized in applicationswhere implementation costs, decoding delays or chip power consumptionissues are paramount. Those skilled in the coding arts shall recognizethat different length block codes derived from different constraintlength convolutional codes having different puncture patterns, differentRS symbol sizes, and different numbers of redundancies, can be used topractice the present inventive coding method and apparatus withoutdeparting from the scope and spirit of the present invention. Thus,these parameters can be varied to obtain a large number of viableconcatenated coding systems useful for different applications.

A number of embodiments of the present invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the presentinvention. For example, the actual implementation of the encoder (anddecoder) described above may be implemented in an integrated circuitdevice, software, firmware, in a combinational logic circuit, Read-OnlyMemory, parallel clocking circuit, or serial circuit as described above.Furthermore, the present inventive method and apparatus can be used invirtually any type of communication system. Its use is not limited to awireless communication system. Alternatively, the present invention canbe used in a wired communication system. Finally, the coding techniquemay be employed at any convenient location within the data communicationsystem. The coder and decoder can reside in both the base stations 106and CPEs 112 of the system of FIG. 1. Accordingly, it is to beunderstood that the invention is not to be limited by the specificillustrated embodiment, but only by the scope of the appended claims.

1. A method of concatenated channel coding of data in a datatransmission system, comprising: (a) obtaining a plurality of dataelements for encoding and transmission in a data transmission system;(b) generating an outer code for the plurality of data elements using an(N,K) Reed-Solomon (RS) code over GF(2^(m)), where N is the length ofthe RS code, K is the message length and m is the length of a symbol,and (c) generating an inner code for the plurality of data elementsusing a (m+1, m) parity-check code; wherein the inner code comprises aplurality of short length block codes derived from a short constraintlength convolutional code using trellis tailbiting.
 2. The method ofconcatenated channel coding as defined in claim 1, wherein the shortconstraint length convolutional code has a constraint length equal toapproximately
 3. 3. The method of concatenated channel coding as definedin claim 1, wherein the short constraint length convolutional code has aconstraint length equal to approximately
 4. 4. The method ofconcatenated channel coding as defined in claim 1, wherein the pluralityof short length block codes are derived from four-state nonsystematic,punctured convolutional codes.
 5. The method of concatenated channelcoding as defined in claim 1, wherein the plurality of short lengthblock codes are derived from four-state nonsystematic, unpuncturedconvolutional codes.
 6. The method of concatenated channel coding asdefined in claim 1, wherein the plurality of short length block codesare derived from four-state nonsystematic, punctured and unpuncturedconvolutional codes.
 7. The method of concatenated channel coding asdefined in claim 1, wherein the inner code has a message length that isa multiple of m.
 8. The method of concatenated channel coding as definedin claim 1, wherein the plurality of short length block codes arederived from an unpunctured rate ½ constraint length 3 convolutionalcode.
 9. The method of concatenated channel coding as defined in claim 8wherein the inner code has a message length equal to or greater than 8bits.
 10. The method of concatenated channel coding as defined in claim8, wherein the inner code has a message length that is a multiple of 8.11. The method of concatenated channel coding as defined in claim 1,wherein the plurality of short length block codes are derived from apunctured rate ⅔ constraint length 3 convolutional code.
 12. The methodof concatenated channel coding as defined in claim 11, wherein the innercode has a message length equal to or greater than 8 bits.
 13. Themethod of concatenated channel coding as defined in claim 11, whereinthe inner code has a message length equal to or greater than 8 bits anda multiple of a puncture period.
 14. The method of concatenated channelcoding as defined in claim 11, wherein the inner code has a messagelength that is a multiple of
 8. 15. The method of concatenated channelcoding as defined in claim 11, wherein the method utilizes a puncturepattern of 11 and
 10. 16. The method of concatenated channel coding asdefined in claim 1, wherein the plurality of short length block codesare derived from an punctured rate ⅘ constraint length 3 convolutionalcode.
 17. The method of concatenated channel coding as defined in claim16, wherein the inner code has a message length equal to or greater than16 bits.
 18. The method of concatenated channel coding as defined inclaim 16, wherein the inner code has a message length equal to orgreater than 16 bits and a multiple of a puncture period.
 19. The methodof concatenated channel coding as defined in claim 16, wherein the innercode has a message length that is a multiple of
 8. 20. The method ofconcatenated channel coding as defined in claim 16; wherein the methodutilizes a puncture pattern of 1001 and
 1110. 21. The method ofconcatenated channel coding as defined in claim 1, further comprising:(d) decoding the plurality of data elements coded using the outer codegenerated in step (b) and the inner code generated in step (c).
 22. Themethod of concatenated channel coding as defined in claim 21, whereinthe decoding step (d) comprises: (1) decoding the inner code generatedin step (c) using a parity-check code decoder, wherein the parity-checkdecoder generates a plurality of m decoded data bits associated andcorresponding to the plurality of data elements obtained in step (a);and (2) decoding the plurality of m decoded data bits of sub-step (1)using an RS code decoder.
 23. The method of concatenated channel codingas defined in claim 22, wherein the parity-check code decoder comprisesa maximum likelihood soft decision parity-check code decoder.
 24. Themethod of concatenated channel coding as defined in claim 1, wherein thecoding steps (b) and (c) utilize a short-depth interleaver.
 25. Themethod of concatenated channel coding as defined in claim 24, whereinthe short-depth interleaver has a depth of
 2. 26. The method ofconcatenated channel coding as defined in claim 24 wherein theshort-depth interleaver has a depth of
 4. 27. The method of concatenatedchannel coding as defined in claim 1, wherein the coding step (c)further comprises generating a bit-permutated inner code.
 28. Aconcatenated channel encoder encoding data in a data transmissionsystem, comprising: (a) means for obtaining a plurality of data elementsfor encoding and transmission in a data transmission system; (b) means,coupled to the data element obtaining means, for generating an outercode for the plurality of data elements using an (N,K) Reed-Solomon (RS)code over GF(2^(m)), where N is the length of the RS code, K is themessage length and m is the length of a symbol; and (c) means, coupledto the outer code generating means, for generating an inner code for theplurality of data elements using a (m+1, m) parity-check code; whereinthe inner code comprises a plurality of short length block codes derivedfrom a short constraint length convolutional code using trellistailbiting.
 29. A concatenated channel encoding apparatus, comprising:(a) an outer code encoder, wherein the outer code encoder generates anouter code for a plurality of data elements using an (N,K) Reed-Solomon(RS) code over GF(2^(m)), where N is the length of the RS code, K is themessage length and m is the length of a symbol, and (b) an inner codeencoder, operatively coupled to the outer code encoder, wherein theinner code encoder generates an inner code for the plurality of dataelements using a (m+1, m) parity-check code, and wherein the inner codecomprises a plurality of short length block codes derived from a shortconstraint length convolutional code using trellis tailbiting.
 30. Adata coder/decoder (CODEC) adapted for use in a data communicationsystem, wherein the data communication system includes a plurality ofcustomer premise equipment (CPE) in communication with associated andcorresponding base stations, and wherein the base stations each includean associated and corresponding media access control (MAC) having aplurality of MAC data messages, and wherein the MAC transports a MACdata message through a MAC data packet that is mapped to at least oneTC/PHY packet in a layered data transport architecture, comprising: (a)an outer code encoder capable of encoding a plurality of data elementsof a selected TC/PHY data packet using an (N,K) Reed-Solomon (RS) codeover GF(2^(m)) where N is the length of the RS code, K is the messagelength and m is the length of a symbol; (b) an inner code encoder,operatively coupled to an output of the outer code encoder, wherein theinner code encoder uses a (m+1, m) parity check code, and wherein theinner code encoder generates a plurality of code words associated withthe plurality of data elements, and wherein the inner code comprises aplurality of short length block codes derived from a short constraintlength convolutional code using trellis tailbiting; and (c) a decoder,operatively coupled to the inner code encoder, wherein the decoderdecodes the code words generated by the inner code encoder.
 31. Anapparatus for coding and decoding data in a data communication system,wherein the data communication system includes a plurality of customerpremise equipment (CPE) in communication with associated andcorresponding base stations, and wherein the base stations each includean associated and corresponding media access control (MAC) having aplurality of MAC data messages, and wherein the MAC transports a MACdata message through a MAC data packet that is mapped to at least oneTC/PHY packet in a layered data transport architecture, comprising: (a)means for generating an outer code for a plurality of data elements of aselected TC/PHY packet using an (N,K) Reed-Solomon (RS) code overGF(2^(m)), where N is the length of the RS code, K is the message lengthand m is the length of a symbol; (b) means, coupled to the outer codegenerating means, for generating an inner code using a (m+1, m) paritycheck code, wherein the inner code comprises a plurality of short lengthblock codes derived from a short constraint length convolutional codeusing trellis tailbiting; and (c) means, coupled to the inner codegenerating means, for decoding the inner code generated by the innercode generating means.
 32. The apparatus as defined in claim 31, whereinthe decoding means comprises: (1) a maximum likelihood soft decisionparity-check code decoding means for decoding the inner code generatedby the inner code generating means, and for generating a plurality ofdecoded bits associated with the plurality of data elements of theselected TC/PHY packet; and (2) a Reed-Solomon code decoding means,operatively coupled to the parity-check code decoding means, fordecoding the plurality of decoded bits generated by the parity-checkcode decoding means.